标题： 随机高斯景观中的 Hessian 特征值分布 显示英文标题

标题： Hessian eigenvalue distribution in a random Gaussian landscape

摘要： 多元宇宙宇宙学的能量景观通常由多维随机高斯势来建模。这类模型的物理预测关键依赖于势能极小值处的Hessian矩阵特征值分布。特别是，真空的稳定性以及慢滚膨胀的动力学对最小特征值的大小非常敏感。之前已经使用鞍点近似法，在$1/N$展开的主阶项中研究了Hessian特征值分布，其中$N$是景观的维度。然而，这种近似对于谱的小特征值端不够充分，因为在该区域次主导项起着重要作用。我们扩展了鞍点方法以考虑次主导贡献。我们还开发了一种新方法，其中特征值分布被确定为随机过程（Dyson布朗运动）终点的平衡分布。在两种方法都适用的情况下，这两种方法的结果是一致的。我们讨论了我们的结果对景观中真空稳定性和慢滚膨胀的影响。 显示英文摘要

摘要： The energy landscape of multiverse cosmology is often modeled by a multi-dimensional random Gaussian potential. The physical predictions of such models crucially depend on the eigenvalue distribution of the Hessian matrix at potential minima. In particular, the stability of vacua and the dynamics of slow-roll inflation are sensitive to the magnitude of the smallest eigenvalues. The Hessian eigenvalue distribution has been studied earlier, using the saddle point approximation, in the leading order of $1/N$ expansion, where $N$ is the dimensionality of the landscape. This approximation, however, is insufficient for the small eigenvalue end of the spectrum, where sub-leading terms play a significant role. We extend the saddle point method to account for the sub-leading contributions. We also develop a new approach, where the eigenvalue distribution is found as an equilibrium distribution at the endpoint of a stochastic process (Dyson Brownian motion). The results of the two approaches are consistent in cases where both methods are applicable. We discuss the implications of our results for vacuum stability and slow-roll inflation in the landscape.